This paper conducted a multi-objective optimization work for a composite internal and film cooling structure. The pitch-to-height ratio of the ribs and inclination angle of the film hole are chosen as the three design variables to enhance the heat transfer performance, improve the film cooling effectiveness and reduce the pressure loss of the internal channel. During the optimization process, Latin hypercube sampling method is adopted to select 26 sample points from the design space. The response values with higher fidelity at the sample points are calculated using CFD simulations. Among the 26 sample points, 21 are used to construct a surrogate model of each objective function while the rest of them are adopted to validate the correctness of the established surrogate model. By combining the Kriging surrogate model with a non-dominated sorting genetic algorithm, the Pareto optimal front is obtained after the optimization process. Finally, comparison and analysis are conducted with respect to the cooling performance and mechanisms between the reference model and the selected three representative optimized models. Results show that the optimized three models can not only improve the film cooling effectiveness, but also reduce the pressure loss of the channel and enhance the heat transfer. In addition, it is found that the optimized model induces an anticlockwise rotating vortex, which entrains more coolant near the target surface. The inclined ribs of the optimized models induce a secondary flow along the inclined ribs, that enhances the flow mixing and augments the heat transfer performance.